{"id":11120,"date":"2023-12-12T22:21:21","date_gmt":"2023-12-13T01:21:21","guid":{"rendered":"https:\/\/numbers-magic.com\/?p=11120"},"modified":"2026-05-19T18:07:04","modified_gmt":"2026-05-19T21:07:04","slug":"block-wise-construction-of-bimagic-squares-multiples-of-orders-49-and-343","status":"publish","type":"post","link":"https:\/\/numbers-magic.com\/?p=11120","title":{"rendered":"Block-Wise Construction of Bimagic Squares of Orders 49 and 343"},"content":{"rendered":"\n<p class=\"has-text-align-right wp-block-paragraph\"><strong><em>Whole the work is done manually on excel sheets<\/em><\/strong>.<\/p>\n\n\n\n<p class=\"has-text-align-justify has-electric-grass-gradient-background has-background wp-block-paragraph\">Below are <strong>bimagic squares<\/strong> written in blocks multiples of orders 7. These are orders 49 and 343. The work is done manually by author in 2011.  It is summarized in the following link. <br><br><strong>Inder J. Taneja<\/strong>, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011,  pp. 1-14, (22.02.2011), h<a href=\"https:\/\/doi.org\/10.48550\/arXiv.1102.3052\">ttps:\/\/doi.org\/10.48550\/arXiv.1102.3052<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Before proceeding further below are the basic formulas to check the sums of <strong>magic<\/strong> and <strong>bimagic<\/strong> squares<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Magic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"176\" height=\"88\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/S-2.png\" alt=\"\" class=\"wp-image-11140\"\/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Bimagic Sum<\/strong><\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" width=\"304\" height=\"93\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2.png\" alt=\"\" class=\"wp-image-11141\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2.png 304w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/Sb-2-300x92.png 300w\" sizes=\"(max-width: 304px) 100vw, 304px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">where <strong><em>n<\/em><\/strong> is the order the magic square.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 49<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In this case, we have<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>49&#215;49<\/sub>:=58849; Sb<sub>49&#215;49<\/sub>:= 94217249<\/strong>, <\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">Blocks of order 7 are pandiagonal and of equal sums, <strong>S<sub>7&#215;7<\/sub> :=8407<\/strong>. The magic squares of order 49 is also pandiagonal. See below the <strong>bimagic square<\/strong> of order 49.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"2764\" height=\"1556\" src=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.png\" alt=\"\" class=\"wp-image-11121\" srcset=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.png 2764w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-300x169.png 300w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-1024x576.png 1024w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-768x432.png 768w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-1600x900.png 1600w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-535x301.png 535w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-1536x865.png 1536w, https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi-2048x1153.png 2048w\" sizes=\"(max-width: 2764px) 100vw, 2764px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\"><strong><em>The work is similar to one studied by Gaston Terry in 1843.<\/em><\/strong><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 54%,rgb(255,105,0) 100%)\">Excel sheet of Bimagic Square of Order 49<\/h4>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-66201883-062e-49fd-a7ae-37e8265eab79\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.xlsx\">49&#215;49-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-66201883-062e-49fd-a7ae-37e8265eab79\">Download<\/a><\/div>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">Bimagic Square of Order 343: Blocks of Orders 7 and 49<\/h3>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\"><strong>Bimagic Square<\/strong> of order 343 with magic and bimagic sums<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><strong>S<sub>343&#215;343<\/sub>:=20176975 and Sb<sub>343&#215;343<\/sub>:= 1582540680175<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">Block of order 49 are <strong>pandiagonal bimagic<\/strong> square with different bimagic sums, <strong>S<sub>49&#215;49<\/sub>:=2882425.<\/strong>  Block of order 7 are also pandiagonal with equal sums <strong>S<sub>7&#215;7<\/sub>:=411775.<\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-justify wp-block-paragraph\">The magic square of order 343 is also pandiagonal. It is tested with software by <a href=\"https:\/\/budshaw.ca\/Download.html\">H. White<\/a>.  <\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(252,185,0) 0%,rgb(255,255,255) 54%,rgb(255,105,0) 100%)\">Excel sheet of Bimagic Square of Order 343<\/h4>\n\n\n\n<div class=\"wp-block-file aligncenter\"><a id=\"wp-block-file--media-2b5cd836-a77c-4ce7-8ee5-69ee2bb1ec4e\" href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/343x343-bi.xlsx\">343&#215;343-bi<\/a><a href=\"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/343x343-bi.xlsx\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-2b5cd836-a77c-4ce7-8ee5-69ee2bb1ec4e\">Download<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-background\" style=\"background:linear-gradient(135deg,rgb(6,147,227) 0%,rgba(69,125,61,0.35) 59%,rgb(155,81,224) 100%)\">References<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Inder J. Taneja<\/strong>, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary 11, 2011, 2011, pp. 1-14, (22.02.2011), h<a href=\"https:\/\/doi.org\/10.48550\/arXiv.1102.3052\">ttps:\/\/doi.org\/10.48550\/arXiv.1102.3052<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11031\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares: Multiples of Orders 8 and 16<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11090\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares Multiples of 25: Orders 25, 125 and 625<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11120\">Block-Wise Construction of Bimagic Squares of Orders 49 and 343.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11129\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares Multiples of 9: Orders 9, 81 and 729<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11142\" target=\"_blank\" rel=\"noreferrer noopener\">Block-Wise Construction of Bimagic Squares of Orders 121 and 1331<\/a>.<\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11168\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 256, 512 and 1024: Blocks of Order 16.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11186\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 200 and 1000: Blocks of Order 8.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11198\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 400, 800, 1600 and 2000: Blocks of Order 16.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=11224\" target=\"_blank\" rel=\"noreferrer noopener\">Bimagic Squares of Orders 100, 110 and 121: Blocks of Orders 10 and 11.<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=13138\">Universal and Upside-Down Magic and Bimagic Squares of Order 16<\/a><\/li>\n\n\n\n<li><strong>Inder J. Taneja<\/strong>,&nbsp;<a href=\"https:\/\/numbers-magic.com\/?p=13336\">Universal and Upside-Down Magic and Bimagic Squares of Order 25<\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Whole the work is done manually on excel sheets. Below are bimagic squares written in blocks multiples of orders 7. These are orders 49 and 343. The work is done manually by author in 2011. It is summarized in the following link. Inder J. Taneja, Bimagic Squares of Bimagic Squares and an Open Problem, Febuarary [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":11121,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-11120","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-days-of-year"],"jetpack_featured_media_url":"https:\/\/numbers-magic.com\/wp-content\/uploads\/2023\/12\/49x49-bi.png","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11120"}],"version-history":[{"count":5,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11120\/revisions"}],"predecessor-version":[{"id":18908,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/posts\/11120\/revisions\/18908"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=\/wp\/v2\/media\/11121"}],"wp:attachment":[{"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11120"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11120"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/numbers-magic.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}